Q5. The amount of sugar (X) contained in 2-kg packets is actually normally distributed with a mean of u = 2.03 kg and a standard deviation of o = 0.014 kg. (a) What proportion of sugar packets are underweight? (b) If an alternative package-filling machine is used for which the weights of the packets are normally distributed with a mean of u = 2.05 kg and a standard deviation of o = 0.016 kg, does this result in an increase or a decrease in the proportion of underweight packets? (c) What is the probability that the amount of sugar of a randomly selected bearing is between 1 and 2 SDs from its mean value? (d) What is the value of c for which there is a 95% probability that a packet weight is within the interval [2.03 – c, 2.03 + c]? (e) What is the value of c for which there is a 95% probability that a packet weight is within the interval [2.5 – c, 2.5 + c]?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![Q5. The amount of sugar (X) contained in 2-kg packets is actually normally
distributed with a mean of u = 2.03 kg and a standard deviation of o
kg.
(a) What proportion of sugar packets are underweight?
(b) If an alternative package-filling machine is used for which the weights of the
packets are normally distributed with a mean of u = 2.05 kg and a standard
deviation of o = 0.016 kg, does this result in an increase or a decrease in the
proportion of underweight packets?
(c) What is the probability that the amount of sugar of a randomly
selected bearing is between 1 and 2 SDs from its mean value?
(d) What is the value of c for which there is a 95% probability that a
packet weight is within the interval [2.03 – c, 2.03 + c]?
(e) What is the value of c for which there is a 95% probability that a
packet weight is within the interval [2.5 – c, 2.5 + c]?
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